Preprint C151/2012
Rational Ergodicity for Skew Products Cylinder Maps

Patricia Romano Cirilo

**Keywords: **
infinite ergodic theory | cylinder skew product | irrational rotation | ergodicity | rationally ergodic | weakly homogeneous.

In this thesis we study the asymptotic behavior of the ergodic Birkhoff Sums for
cylinder skew products over irrational rotation preserving a σ-finite measure. We prove
that such maps are ergodic, rationally ergodic and weakly homogeneous, calculating
explicitly the Ergodic Sums for an increasing sequence of time and identifying the return
sequence. From that, it is possible to obtain a second order ergodic theorem, which asserts
that the double average renormalized by the return sequence converges to the integral of
the observable function almost everywhere. We recall that the classical Birkhoff Theorem
does not hold when the invariant measure is infinite.